 Welcome back to our lecture series math 1060 trigonometry for students at Southern Thai University as usual I'll be your professor today. Dr. Andrew misaligned chapter 8 Which we begin here in lecture 24. It's gonna be about the topic of oblique triangles. What do we mean by that? well in this in this trigonometry course very early on we studied triangles specifically in our lecture series chapter 2 was focused on solving right triangles and chapter 8 we're gonna return to the study of triangles But up to this point. We've essentially only studied right triangles. There was an occasional rare I should say equilateral or a Saucely's triangle. Let it pop up here or there, but for the most part Almost exclusively we've been studying right triangles in this chapter. We're gonna study Non-right triangles which are called these oblique triangles, so we don't have a right angle In the triangle anymore, but we want to solve it and remember what we mean by solving here Like we did back in chapter 2 is we're gonna have some Combination of angles that are known so like maybe we know angles A and B and we know some combination of sides We're always gonna have three of these Three angles three sides are a combination of three and then we want to determine the missing pieces That are still left to be determined, right? So if we know angles B and A and B, could we find out angle C? Can we find the side lengths B and C? And so it turns out the techniques We we've learned when it came to right triangles do have some application to those oblique triangles as well When it came to right triangles, we did things like so katoa sine cosine tangent to help us determine This missing information for oblique triangles We have to be a little bit more careful because the trig functions like sine cosine tangent don't apply directly to oblique triangles But using things like the law of sines law of cosines which will develop in the in these in these lectures right now in this chapter We'll be able to do that very very quickly Now you'll see these diagrams a lot one like this a few things I want to caution you about is one never assume that the diagrams I draw are drawn to scale because honestly I make no effort to draw these to scale You'll see this exact same picture over and over and over again because it's just copy and pasted on these slides there So don't assume things are drawn to scale. So don't panic about things like that Also, just remember it's convention when we draw triangles here It's quite typical when referring to the vertices of a triangle. We use a capital Roman letter like abc When it comes to a triangle referring to the vertex these corners here also indicates an angle And so we commonly use the exact same capital Roman letter to refer to the measure of the angle as well So you have angle a angle B angle C, etc. All right Opposite any angle will be a side of the triangle. So this would be the side which is opposite of angle a It's common convention to use the same Roman letter But lowercase to also indicate the opposite side of the angle there So capital a and lowercase a are an angle and its opposite side Likewise, you have angle B with side B and this represents another angle opposite side We say this a lot. So we often use abbreviations acronyms to refer to this This is an AOS an angle opposite side whenever you have an angle opposite side pair That's actually really good when it comes to solving for these oblique triangles whenever you have an AOS That's when the loft signs comes into play, but we'll talk more about that in the upcoming video Okay, so like I said when it comes to solving oblique triangles, you're always gonna have some information about the triangle And then we have to solve for the remaining information in the triangle So you need at least three bits of information in order to solve a triangle if you have less than three like if you only know two Two of the bits information. So this is two of the angles or sides or some combination in angle and a side That's not enough to solve the triangle uniquely So you need at least three if you have if you have three in most cases You can then solve the remaining information the triangle and so you're gonna see these acronyms here So a a a a a a s a s a what these are? There's just abbreviations here in which case every time you see an angle Excuse me every time you see an a that'll refer to this symbol angle whenever you see an s that'll refer to side And so what you want when you what this means for us is that if you see something like a a s You should interpret as you know one angle up, you know a second angle And you know a side of the triangle and the the order of these things does matter You have an a a s versus an a s a the difference here is that with a s a the side You know is between the two angles, you know on the other hand if you know a a s This means the side you know is not between the two angles In fact the side would be opposite this angle right here And so you have an a os as opposed to a s a which you don't have an a os You actually have the interior side That's between the two angles and so the order of these symbols it tells you This is just a code to help us know what initial information do we know about the triangle? How can we solve it like so? Okay, so the first one which we've actually studied this one before we're not going to see much more of a a a In this chapter for the following reason a a a means you know the three angles of the triangle And it turns out there is no method to solve a triangle if you Know the three angles of the triangle and it turns out that this a a a case cannot be solved because there's not unique triangles a a a actually describes similar triangles that we've talked about before similar triangles Similar triangles occur exactly when two triangles Have the same three exact Angles and in fact given any three angles a a a there's an infinite number of single similar triangles that'll share those Same angles and so the best we can say about a a a the best we can say about similar triangles Is that this the three unknown sides will be proportional to each other? But they're not necessarily unique to each other So a a a is not a congruent condition for triangles You only get similar triangles and so what I mean by congruent conditions Is recall that two triangles are congruent to each other if there's some correspondence between angles and sides So that the three angles are congruent and the three sides are congruent to each other If you know that two triangles have the same three angles that shows they're similar triangles Doesn't give you congruence. We don't know the triangles are congruent to each other But these other conditions can actually guarantee congruence of triangles So there's the a a s condition angle angle side It turns out if you know two of the angles of a triangle and you know A side that's not interior to those two angles Then you can solve for the entire triangle using the law of signs We'll see how to do that later on in this lecture And this actually shows us then that if two triangles have the same two angles and exterior side Then those two triangles have to be congruent to each other because we can find the missing pieces and they'd have to be the same Similar to the a the a s condition is the a s a condition angle side angle So the side is now interior to the two angles It turns out that using the law of signs you can show that if you know two angles and the interior side Then you can solve for the triangle So whether whether the side is exterior or interior you can solve for the the missing information using the law of signs Although the approach is a little bit different, which is why we still need to separate these cases So if two triangles Satisfy the same a s a condition. They're actually congruent to each other We're going to see that in this lecture here In the next lecture, we're going to pursue two more cases a are excuse me s a s and s s s That is the side angle side condition and the side side side condition Side side side means we know we know all three sides of the triangle And then using the law of cosigns, which we'll introduce in the next lecture We can show that we can actually determine what the three angles have to be And so if two triangles have the same three side side sides Then actually the two triangles have to be congruent to each other Similar to the side side side condition is the side angle side condition for which what we mean here is that we have two sides And the angle is interior the angles between the two sides If we have side angle side, then the law of cosigns will apply And then we can find the missing sides of the triangle So if two triangles satisfy the same side angle side condition, they have to be congruent to each other The last the last possibility on this list is the side side angle condition, which means So s s a so in s s a what we have here is we know two sides of the triangle and we know an angle But the angle is exterior to the two We're going to refer to this as the ambiguous case This case is a little bit more complicated and we'll have to be treated in its own right Which we will do that in its own lecture following the lectures about lost signs and law of cosigns So in this lecture 24 the subsequent videos will learn about the law of signs And how we can use it to solve the a s and the a s a condition using angle opposite sides a os In lecture 25 we'll talk about the law of cosigns and show us how to solve the s a s and the s s s condition And then in lecture 26 we'll consider the ambiguous case s s a all by itself